
where \( S_{\text{path}} ({\bf x}(t\rightarrow t_f))\)  is a corresponding action and \(\mathcal {D}({\bf x})\sim \prod _{i=1}^n d{\bf x}_i\)  is the integral measure over all the trajectories that start at \({\bf x}_0 = {\bf x}(0)\) . Numerically, this formulation provides a way to generate evolutionary trajectories under artificial selection as an exponentially weighted ensemble from trajectories under natural selection [1]}, [2]}. Moreover,
if \(\lambda \)  is small, the integral is dominated by the trajectories that are close to the most likely (classical) trajectory \(\hat{{\bf x}}(t\rightarrow t_f)\) , and the path integral can be approximated using the semi-classical method; see ref. [1]}.
